is known as the Helstrom matrix [1]}. Now, the error probability
\(P_{e}\)  has to be minimized over \(E_{B}\) . Since \(\hbox{Tr}[\Lambda ]=0\) , the
Helstrom matrix has both positive and negative eigenvalues and the minimum
\(P_{e}\)  is attained if \(E_{B}\)  is chosen as the projector over \(\Lambda _{+}\) ,
i.e., the positive subspace of \(\Lambda \) . Assuming this optimal operator we
have \(\hbox{Tr}[E_{B}\Lambda ]=\mathop {\text{Tr}}\nolimits [\Lambda _{+}]=\frac{1}{2}\mathop {\text{Tr}}\nolimits |\Lambda |\)  with \(|\Lambda |=\sqrt{\Lambda ^{\dagger }\Lambda }\) . Thus the minimal error probability is
given by
\(P_{e}=\frac{1}{2}\left[ 1-T(\rho _{A}^{M},\rho _{B}^{M})\right] ~,\) 
